A Characteristic of IBL Teaching: Mediating the Interaction Between Students and Mathematics

A question that comes up frequently is, "What are some of the main features of IBL teaching?"  One of the main characteristics of IBL teaching that I'd like to highlight in this post is mediating the interaction between students and mathematics.  

Education is full of monikers for active, student-centered teaching.  We say things like, "Guide on the side," "Mentor in the middle," "Coach," etc.  These are nice ways to think about the nature of IBL teaching, and it's important to unpack what these things specifically mean.

In IBL Math there exists a particular role for instructors, which is to set-up and mediate the problem-solving process.  Students are engaged in exploring the mathematical landscape via well-chosen, logically ordered sequences of problems.  As students engage in their explorations, the instructor works to support the class so that students' interactions with the mathematics results in learning.  

Here are some ways IBL instructors mediate the interaction between students and mathematics.  Instructors

1. design/adapt appropriate curriculum.  The problem sequences must be matched to the goals of the course, must be logically consistent and coherent, and meet the needs of the specific students in the class.
2. keep students going without getting overly stuck.  That is, struggle is good if the struggle is fruitful.  Being stuck is part of the learning process, and students come in with a "tolerance threshold" for being stuck.  It's important to stay within these tolerances so that students stay in the game.  One positive outcome of an IBL course is that tolerance for being stuck can increase. It's something that can be improved upon.
3. provide scaffolding.  This idea is related to the previous point is being able to control the "being stuck phases" by offering "bread crumbs" (hints, lemmas, sub problems, deploying group work, etc.) in just the right type and size to keep the cognitive level of the task high, while simultaneously avoiding student shutdown due to being overly stuck.  This is where calibrating to specific classes comes into play, and making good choices depends on having good data about students.  Listening, observing, and interacting regularly with students forms the foundation of this piece.  With good data in hand, it's often clear what scaffolding is needed.  "Oh, students are having trouble thinking about supremum.  Students may need to work on a couple of problems about upper bounds…"
4. set the class structure and environment for positive discourse.  Class presentations are of no value if the class environment is not setup appropriately.  IBL instructors set the roles, expectations, and procedures of class discourse, which is used to share and validate ideas.  The ethos of the class must include inquiry, discovery, mutual respect for students and learning, and valuing "mistakes" as important discoveries.
5. provide structure of the material being covered, so that students know where they are in the mathematical landscape.  This is an excellent place to inject small lectures and/or organizational tasks, where students are not necessarily solving new problems, but organizing the information they have recently studied.  

The central focus is on the math and how students are working on the math.  The first major component is creating problems that guide students through the mathematical landscape.  Deploying these tasks determines the nature of the interaction students have with the mathematics.   As students work on solving problems, help or direction is needed.  The instructor provides enough assistance to keep students in the learning zone, without taking away all of the fun and exploration.  When students make progress and discoveries, the instructor provides a forum for the ideas to be shared, vetted, and learned by all.

This is the notion of mediating the interaction between students and the mathematics. It is a central component of effective IBL teaching, and a healthy perspective for instructors to embrace.

Related Posts:

IBL Teaching Levels

Asking Good Questions

Sense-Making Continental Divide

The Sense-Making Continental Divide

Frequently I am involved in good discussions about what is IBL, whether it is the strict Moore Method or something else.   This is a good topic for discussion, and I'd like to share my thoughts on the issue.   I'm going to approach this topic with a simple, but useful model that highlights a major structural component of IBL instruction.

Math courses are taught in a variety of ways, and even within the IBL community there exists numerous differences.  This is something that should be expected, because environments and goals differ across institutions.  Just as we would not expect Michelin star restaurants to be identical across the world, we should expect that successful instruction will be different and varied to suit the needs across different institutions.  This honors the diversity of humanity, and allows a teacher to be true to her or his personality

Here's the model I have in mind:

In this model I use an idea I call the "Sense-Making Continental Divide."  A key feature that defines IBL instruction is that students regularly are encouraged to do the sense-making tasks, including validation of solutions or proofs, understanding statements of problems, and working from definitions and first principles.  IBL instructors set up courses to get students over the Sense-Making Continental Divide, where students are regularly doing activities that require students to think, decide, explain, evaluate, and reflect.  You have crossed over the SMCD if your students are (a) deeply engaged in rich mathematics and (b) have opportunities to collaborate and discuss ideas and solutions.  (This succinct characterization of IBL is from Sandra Laursen, University of Colorado Boulder.)

It's easy to see there are numerous options for implementing sense-making activities.  From the palette of teaching options is derived the multiple variations of IBL.  In my perspective, this explains why IBL comes in so many different forms.  We have more choices, and the "correct" teaching decision depends on real-time conditions in the specific class setting an instructor is in.

What typifies traditional instruction is that the instructor does the processing and sense-making through presentations.  "This is the proof of theorem 3.6..."  Students do not get many opportunities in class to do the structuring or validation.  In such classes, students might be unintentionally encouraged to memorize facts rather than make sense of the ideas.

The Hybrid IBL zone contains the different forms of IBL methods that are often a result of practical limitations instructors face.  There may be a required syllabus, or a course may be predominantly procedural in nature (e.g. calculus). A course may have large enrollment, or an instructor may not have the requisite skills or experience to comfortably run a full IBL course.

It could also be the case that the (full) IBL class instructor shares a solution on occasion in the event that students are floundering, and moving ahead would be more beneficial mathematically for the students.  Flexibility and adaptability are key traits of effective instruction.  An IBL course may change during the term to adapt to specific needs.

Is your course an IBL course of some kind?  One way to see is if your students are regularly over the Sense-Making continental Divide.

For more about your personal IBLishness, see also this post on IBL Levels.

IBL Levels (with apologies to Van Hiele)

In this post I present the IBL levels, a la the Van Hiele levels, which sincere apologies to Van Hiele. These are based on my observations of IBL instructors over the past decade.

A very important point is that the IBL levels presented here are simplistic (reductionist) and are intended as a form of guidance to instructors.  The IBL levels indicate the direction in which you ought to go and provide a framework for evaluating your own teaching.

Level 0: non-IBL, teacher-centered instruction.


Level 1: some student engagement via occasional "active lecture" methods like
Think-Pair-Share, concept questions, or students working out examples.  The instructor and textbook remain as the predominant mathematical authorities, where students seek confirmation from the instructor or textbook to check if their answers are correct.


Level 2: a significant percentage of class time (50% or more) is used to engage students in solving problems, discussing solutions, and peer reviewing work.  The problem solving activities are focused on problems that students do not know the answer to or are not shown the strategy or solution method.  Students are supported and encouraged to make their own conclusions, think for themselves, and share their insights.  Students have a predilection for determining the correctness of solutions without always seeking external validation from an instructor or book.  Students have frequent opportunities to engage in rich mathematical task.


Level 3: full IBL.

While level 3 is a lofty goal to set for oneself for teaching all courses, level 2 is more easily attainable, especially for courses like freshman calculus.  Levels 2 and 3 are then goals that can be achieved by math instructors.  In particular, level 2 does not require wholesale changes in curriculum, and is thus appropriate for various contexts and environments.  It should be noted, however, that level 2 does not have the same potential as level 3 for transformative experiences.  There are always tradeoffs.

For novice IBL instructors, you do not have to wait for the "right course" to begin your journey in IBL instruction.  Level 1 can be done in any class, and allows you to test the waters and build up your own IBL teaching skills and understanding of IBL methodology.  Making the transition to level 2 would be much easier, and it sets the foundation for making the transition to full IBL instruction.

If you have been hesitating, start with level one and try out Think Pair Share!